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The graph of a function $ f $ and its tangent line at $ 0 $ are shown. What is the value of $ \displaystyle \lim_{x\to 0} \frac{f(x)}{e^x - 1} $?

$\lim _{z \rightarrow 0} \frac{f^{\prime}(x)}{e^{x}=1}$

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Missouri State University

Oregon State University

Harvey Mudd College

Boston College

So we're given a graph and a function and we want to determine its limit. So we see that when X equals zero, we end up getting 0/0. And that's an indeterminant form. So instead what we do is we take the derivative of the numerator, so f prime of X and then the derivative of the denominator, which would just give us me to the X. And then we take the limit as x approaches zero on both the top and the bottom. As a result of that, we see that the limit as X approaches zero of the numerator is going to give us one. And then when X approaches zero and the denominator, the limit of that is going to be one based on that. Our final answer will be one. Um And we know that the numerator is zero because we see that uh with the graph as X approaches zero, the graph itself will be zero. So that is why we get the indeterminate form in the first place, and then we see the E to the zero is 1 to -1, it's going to be zero. Which is why we get 0/0 as our indeterminant form, um which allowed us to use the hotel's rules.

California Baptist University